4.3 Partial derivatives (Page 4/11)

Calculus volume 3 Page 4 / 11

We can calculate a partial derivative of a function of three variables using the same idea we used for a function of two variables. For example, if we have a function $f$ of $x, y, and z,$ and we wish to calculate $\partial f / \partial x,$ then we treat the other two independent variables as if they are constants, then differentiate with respect to $x .$

Calculating partial derivatives for a function of three variables

Use the limit definition of partial derivatives to calculate $\partial f / \partial x$ for the function

f (x, y, z) = x^{2} - 3 x y + 2 y^{2} - 4 x z + 5 y z^{2} - 12 x + 4 y - 3 z .

Then, find $\partial f / \partial y$ and $\partial f / \partial z$ by setting the other two variables constant and differentiating accordingly.

We first calculate $\partial f / \partial x$ using [link] , then we calculate the other two partial derivatives by holding the remaining variables constant. To use the equation to find $\partial f / \partial x,$ we first need to calculate $f (x + h, y, z) :$

\begin{matrix} f (x + h, y, z) & = {(x + h)}^{2} - 3 (x + h) y + 2 y^{2} - 4 (x + h) z + 5 y z^{2} - 12 (x + h) + 4 y - 3 z \\ = x^{2} + 2 x h + h^{2} - 3 x y - 3 x h + 2 y^{2} - 4 x z - 4 h z + 5 y z^{2} - 12 x - 12 h + 4 y - 3 z \end{matrix}

and recall that $f (x, y, z) = x^{2} - 3 x y + 2 y^{2} - 4 z x + 5 y z^{2} - 12 x + 4 y - 3 z .$ Next, we substitute these two expressions into the equation:

\begin{matrix} \frac{\partial f}{\partial x} & = lim_{h \to 0} [\frac{x^{2} + 2 x h + h^{2} - 3 x y - 3 h y + 2 y^{2} - 4 x z - 4 h z + 5 y z^{2} - 12 x - 12 h + 4 y - 3 z}{h} \\ - \frac{x^{2} - 3 x y + 2 y^{2} - 4 x z + 5 y z^{2} - 12 x + 4 y - 3 z}{h}] \\ = lim_{h \to 0} [\frac{2 x h + h^{2} - 3 h y - 4 h z - 12 h}{h}] \\ = lim_{h \to 0} [\frac{h (2 x + h - 3 y - 4 z - 12)}{h}] \\ = lim_{h \to 0} (2 x + h - 3 y - 4 z - 12) \\ = 2 x - 3 y - 4 z - 12. \end{matrix}

Then we find $\partial f / \partial y$ by holding $x and z$ constant. Therefore, any term that does not include the variable $y$ is constant, and its derivative is zero. We can apply the sum, difference, and power rules for functions of one variable:

\begin{matrix} \frac{\partial}{\partial y} [x^{2} - 3 x y + 2 y^{2} - 4 x z + 5 y z^{2} - 12 x + 4 y - 3 z] \\ = \frac{\partial}{\partial y} [x^{2}] - \frac{\partial}{\partial y} [3 x y] + \frac{\partial}{\partial y} [2 y^{2}] - \frac{\partial}{\partial y} [4 x z] + \frac{\partial}{\partial y} [5 y z^{2}] - \frac{\partial}{\partial y} [12 x] + \frac{\partial}{\partial y} [4 y] - \frac{\partial}{\partial y} [3 z] \\ = 0 - 3 x + 4 y - 0 + 5 z^{2} - 0 + 4 - 0 \\ = −3 x + 4 y + 5 z^{2} + 4. \end{matrix}

To calculate $\partial f / \partial z,$ we hold x and y constant and apply the sum, difference, and power rules for functions of one variable:

\begin{matrix} \frac{\partial}{\partial z} [x^{2} - 3 x y + 2 y^{2} - 4 x z + 5 y z^{2} - 12 x + 4 y - 3 z] \\ = \frac{\partial}{\partial z} [x^{2}] - \frac{\partial}{\partial z} [3 x y] + \frac{\partial}{\partial z} [2 y^{2}] - \frac{\partial}{\partial z} [4 x z] + \frac{\partial}{\partial z} [5 y z^{2}] - \frac{\partial}{\partial z} [12 x] + \frac{\partial}{\partial z} [4 y] - \frac{\partial}{\partial z} [3 z] \\ = 0 - 0 + 0 - 4 x + 10 y z - 0 + 0 - 3 \\ = −4 x + 10 y z - 3. \end{matrix}

Got questions? Get instant answers now!

Use the limit definition of partial derivatives to calculate $\partial f / \partial x$ for the function

f (x, y, z) = 2 x^{2} - 4 x^{2} y + 2 y^{2} + 5 x z^{2} - 6 x + 3 z - 8 .

Then find $\partial f / \partial y$ and $\partial f / \partial z$ by setting the other two variables constant and differentiating accordingly.

$\frac{\partial f}{\partial x} = 4 x - 8 x y + 5 z^{2} - 6, \frac{\partial f}{\partial y} = −4 x^{2} + 4 y, \frac{\partial f}{\partial z} = 10 x z + 3$

Got questions? Get instant answers now!

Calculating partial derivatives for a function of three variables

Calculate the three partial derivatives of the following functions.

$f (x, y, z) = \frac{x^{2} y - 4 x z + y^{2}}{x - 3 y z}$
$g (x, y, z) = sin (x^{2} y - z) + cos (x^{2} - y z)$

In each case, treat all variables as constants except the one whose partial derivative you are calculating.

$\begin{matrix} \frac{\partial f}{\partial x} & = \frac{\partial}{\partial x} [\frac{x^{2} y - 4 x z + y^{2}}{x - 3 y z}] \\ = \frac{\frac{\partial}{\partial x} (x^{2} y - 4 x z + y^{2}) (x - 3 y z) - (x^{2} y - 4 x z + y^{2}) \frac{\partial}{\partial x} (x - 3 y z)}{{(x - 3 y z)}^{2}} \\ = \frac{(2 x y - 4 z) (x - 3 y z) - (x^{2} y - 4 x z + y^{2}) (1)}{{(x - 3 y z)}^{2}} \\ = \frac{2 x^{2} y - 6 x y^{2} z - 4 x z + 12 y z^{2} - x^{2} y + 4 x z - y^{2}}{{(x - 3 y z)}^{2}} \\ = \frac{x^{2} y - 6 x y^{2} z - 4 x z + 12 y z^{2} + 4 x z - y^{2}}{{(x - 3 y z)}^{2}} \end{matrix}$
$\begin{matrix} \frac{\partial f}{\partial y} & = \frac{\partial}{\partial y} [\frac{x^{2} y - 4 x z + y^{2}}{x - 3 y z}] \\ = \frac{\frac{\partial}{\partial y} (x^{2} y - 4 x z + y^{2}) (x - 3 y z) - (x^{2} y - 4 x z + y^{2}) \frac{\partial}{\partial y} (x - 3 y z)}{{(x - 3 y z)}^{2}} \\ = \frac{(x^{2} + 2 y) (x - 3 y z) - (x^{2} y - 4 x z + y^{2}) (−3 z)}{{(x - 3 y z)}^{2}} \\ = \frac{x^{3} - 3 x^{2} y z + 2 x y - 6 y^{2} z + 3 x^{2} y z - 12 x z^{2} + 3 y^{2} z}{{(x - 3 y z)}^{2}} \\ = \frac{x^{3} + 2 x y - 3 y^{2} z - 12 x z^{2}}{{(x - 3 y z)}^{2}} \end{matrix}$
$\begin{matrix} \frac{\partial f}{\partial z} & = \frac{\partial}{\partial z} [\frac{x^{2} y - 4 x z + y^{2}}{x - 3 y z}] \\ = \frac{\frac{\partial}{\partial z} (x^{2} y - 4 x z + y^{2}) (x - 3 y z) - (x^{2} y - 4 x z + y^{2}) \frac{\partial}{\partial z} (x - 3 y z)}{{(x - 3 y z)}^{2}} \\ = \frac{(−4 x) (x - 3 y z) - (x^{2} y - 4 x z + y^{2}) (−3 y)}{{(x - 3 y z)}^{2}} \\ = \frac{−4 x^{2} + 12 x y z + 3 x^{2} y^{2} - 12 x y z + 3 y^{3}}{{(x - 3 y z)}^{2}} \\ = \frac{−4 x^{2} + 3 x^{2} y^{2} + 3 y^{3}}{{(x - 3 y z)}^{2}} \end{matrix}$
$\begin{matrix} \frac{\partial f}{\partial x} & = \frac{\partial}{\partial x} [sin (x^{2} y - z) + cos (x^{2} - y z)] \\ = (cos (x^{2} y - z)) \frac{\partial}{\partial x} (x^{2} y - z) - (sin (x^{2} - y z)) \frac{\partial}{\partial x} (x^{2} - y z) \\ = 2 x y cos (x^{2} y - z) - 2 x sin (x^{2} - y z) \\ \frac{\partial f}{\partial y} & = \frac{\partial}{\partial y} [sin (x^{2} y - z) + cos (x^{2} - y z)] \\ = (cos (x^{2} y - z)) \frac{\partial}{\partial y} (x^{2} y - z) - (sin (x^{2} - y z)) \frac{\partial}{\partial y} (x^{2} - y z) \\ = x^{2} cos (x^{2} y - z) + z sin (x^{2} - y z) \\ \frac{\partial f}{\partial z} & = \frac{\partial}{\partial z} [sin (x^{2} y - z) + cos (x^{2} - y z)] \\ = (cos (x^{2} y - z)) \frac{\partial}{\partial z} (x^{2} y - z) - (sin (x^{2} - y z)) \frac{\partial}{\partial z} (x^{2} - y z) \\ = − cos (x^{2} y - z) + y sin (x^{2} - y z) \end{matrix}$

Got questions? Get instant answers now!

<< Chapter < Page Page > Chapter >>

Practice Key Terms 4

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

Source: OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 3' conversation and receive update notifications?

Ask

©flickr: Tim	Hazard Quiz By Royalle Moore Start Quiz
	13 Sociology 13 Aging and the Elderly MCQ By OpenStax Start Quiz
	Are you a 5sos fan? By Rylee Minllic Start Quiz
©flickr: Gage	How well do you know Ross Lynch? By Brianna Beck Start Quiz
	10 Sociology 10 Global Inequality MCQ By OpenStax Start Quiz
	15 Dr. Amberg Pharm quiz By Brooke Delaney Start Exam
	Statistics Final Review By Madison Christian Start Exam
	12 Sociology 12 Gender, Sex, and Sexuality MCQ By OpenStax Start Quiz
	College physics for ap® courses By OpenStax Read Online Course
©flickr: quinn	Neuroanatomy By George Turner Start Quiz